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Distribution of periodic orbits of symbolic and axiom A flows. (English) Zbl 0637.58013
Let A be an $$n\times n$$ 0-1 matrix. For given matrix A, $$\Sigma_{\hat A}$$ denotes the set of all doubly infinite sequences and $$C(\Sigma_ A)$$ the space of continuous complex-valued functions on $$\Sigma_ A$$. For $$f\in C(\Sigma_{\hat A})$$ let $$(\Sigma^ f_{\hat A},\sigma^ f)$$ denote the symbolic flow, defined on the suspension space $$\Sigma^ f_ A$$ by usual manner. Let G be a bounded Borel measurable real-valued function on $$\Sigma^ f_ A$$. Periodic orbits of $$(\Sigma^ f_ A,\sigma^ f)$$ are denoted by $$\tau$$, $$\tau$$ (G) denotes the integral of G, with respect to time, over one period for $$\tau$$. Thus $$\tau$$ (1) is the period of $$\tau$$, and $$\tau$$ (G)/$$\tau$$ (1) is the mean value of G over $$\tau$$. Define measure $$Q^ f$$ on $${\mathbb{R}}^+$$ by $$Q^ f(I)=\#\{\tau: \tau (1)\in I\}$$. The main results of this paper concern the asymptotic behavior of this measure, when the flow is weakly mixing. Also further analogous measures are constructed and investigated.
Reviewer: A.Klíč

##### MSC:
 37A99 Ergodic theory 37D15 Morse-Smale systems 28D99 Measure-theoretic ergodic theory
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